T-symmetry can be shown to be equivalent to the conservation of entropy, by Noether's Theorem. But as the second law of thermodynamics does not permit entropy to be conserved in general, it follows that the observable universe does not in general show symmetry under time reversal. In other words, time is said to be non-symmetric, or asymmetric, except for special equilibrium states when the second law of thermodynamics predicts the time symmetry to hold.
However, quantum noninvasive measurements are predicted to violate time symmetry even in equilibrium,[1] contrary to their classical counterparts, although this has not yet been experimentally confirmed.

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A toy called the teeter-totter illustrates, in cross-section, the two aspects of time reversal invariance. When set into motion atop a pedestal (rocking side to side, as in the image), the figure oscillates for a very long time. The toy is engineered to minimize friction and illustrate the reversibility of Newton's laws of motion. However, the mechanically stable state of the toy is when the figure falls down from the pedestal into one of arbitrarily many positions. This is an illustration of the law of increase of entropy through Boltzmann's identification of the logarithm of the number of states with the entropy.

Physicists also discuss the time-reversal invariance of local and/or macroscopic descriptions of physical systems, independent of the invariance of the underlying microscopic physical laws.
For example, Maxwell's equations with material absorption or Newtonian mechanics with friction are not time-reversal invariant at the macroscopic level where they are normally applied, even if they are invariant at the microscopic level; when one includes the atomic motions, the "lost" energy is translated into heat.

Our daily experience shows that T-symmetry does not hold for the behavior of bulk materials. Of these macroscopic laws, most notable is the second law of thermodynamics. Many other phenomena, such as the relative motion of bodies with friction, or viscous motion of fluids, reduce to this, because the underlying mechanism is the dissipation of usable energy (for example, kinetic energy) into heat.

The question of whether this time-asymmetric dissipation is really inevitable has been considered by many physicists, often in the context of Maxwell's demon. The name comes from a thought experiment described by James Clerk Maxwell in which a microscopic demon guards a gate between two halves of a room. It only lets slow molecules into one half, only fast ones into the other. By eventually making one side of the room cooler than before and the other hotter, it seems to reduce the entropy of the room, and reverse the arrow of time. Many analyses have been made of this; all show that when the entropy of room and demon are taken together, this total entropy does increase. Modern analyses of this problem have taken into account Claude E. Shannon's relation between entropy and information. Many interesting results in modern computing are closely related to this problem — reversible computing, quantum computing and physical limits to computing, are examples. These seemingly metaphysical questions are today, in these ways, slowly being converted into hypotheses of the physical sciences.

The current consensus hinges upon the Boltzmann-Shannon identification of the logarithm of phase space volume with the negative of Shannon information, and hence to entropy. In this notion, a fixed initial state of a macroscopic system corresponds to relatively low entropy because the coordinates of the molecules of the body are constrained. As the system evolves in the presence of dissipation, the molecular coordinates can move into larger volumes of phase space, becoming more uncertain, and thus leading to increase in entropy.

One can, however, equally well imagine a state of the universe in which the motions of all of the particles at one instant were the reverse (strictly, the CPT reverse). Such a state would then evolve in reverse, so presumably entropy would decrease (Loschmidt's paradox). Why is 'our' state preferred over the other?

One position is to say that the constant increase of entropy we observe happens only because of the initial state of our universe. Other possible states of the universe (for example, a universe at heat death equilibrium) would actually result in no increase of entropy. In this view, the apparent T-asymmetry of our universe is a problem in cosmology: why did the universe start with a low entropy? This view, if it remains viable in the light of future cosmological observation, would connect this problem to one of the big open questions beyond the reach of today's physics — the question of initial conditions of the universe.

An object can cross through the event horizon of a black hole from the outside, and then fall rapidly to the central region where our understanding of physics breaks down. Since within a black hole the forward light-cone is directed towards the center and the backward light-cone is directed outward, it is not even possible to define time-reversal in the usual manner. The only way anything can escape from a black hole is as Hawking radiation.

The time reversal of a black hole would be a hypothetical object known as a white hole. From the outside they appear similar. While a black hole has a beginning and is inescapable, a white hole has an ending and cannot be entered. The forward light-cones of a white hole are directed outward; and its backward light-cones are directed towards the center.

The event horizon of a black hole may be thought of as a surface moving outward at the local speed of light and is just on the edge between escaping and falling back. The event horizon of a white hole is a surface moving inward at the local speed of light and is just on the edge between being swept outward and succeeding in reaching the center. They are two different kinds of horizons—the horizon of a white hole is like the horizon of a black hole turned inside-out.

Since most systems are asymmetric under time reversal, it is interesting to ask whether there are phenomena that do have this symmetry. In classical mechanics, a velocity v reverses under the operation of T, but an acceleration does not. Therefore, one models dissipative phenomena through terms that are odd in
v. However, delicate experiments in which known sources of dissipation are removed reveal that the laws of mechanics are time reversal invariant. Dissipation itself is originated in the second law of thermodynamics.

The motion of a charged body in a magnetic field, B involves the velocity through the Lorentz force term v×B, and might seem at first to be asymmetric under T. A closer look assures us that B also changes sign under time reversal. This happens because a magnetic field is produced by an electric current, J, which reverses sign under T. Thus, the motion of classical charged particles in electromagnetic fields is also time reversal invariant. (Despite this, it is still useful to consider the time-reversal non-invariance in a local sense when the external field is held fixed, as when the magneto-optic effect is analyzed. This allows one to analyze the conditions under which optical phenomena that locally break time-reversal, such as Faraday isolators and directional dichroism, can occur.) The laws of gravity also seem to be time reversal invariant in classical mechanics.

In physics one separates the laws of motion, called kinematics, from the laws of force, called dynamics. Following the classical kinematics of Newton's laws of motion, the kinematics of quantum mechanics is built in such a way that it presupposes nothing about the time reversal symmetry of the dynamics. In other words, if the dynamics are invariant, then the kinematics will allow it to remain invariant; if the dynamics is not, then the kinematics will also show this. The structure of the quantum laws of motion are richer, and we examine these next.

Two-dimensional representations of parity are given by a pair of quantum states that go into each other under parity. However, this representation can always be reduced to linear combinations of states, each of which is either even or odd under parity. One says that all irreducible representations of parity are one-dimensional. Kramers' theorem states that time reversal need not have this property because it is represented by an anti-unitary operator.

This section contains a discussion of the three most important properties of time reversal in quantum mechanics; chiefly,

that it has two-dimensional representations with the property T2 = −1.

The strangeness of this result is clear if one compares it with parity. If parity transforms a pair of quantum states into each other, then the sum and difference of these two basis states are states of good parity. Time reversal does not behave like this. It seems to violate the theorem that all abelian groups be represented by one-dimensional irreducible representations. The reason it does this is that it is represented by an anti-unitary operator. It thus opens the way to spinors in quantum mechanics.

Eugene Wigner showed that a symmetry operation S of a Hamiltonian is represented, in quantum mechanics either by a unitary operator, S = U, or an antiunitary one, S = UK where U is unitary, and K denotes complex conjugation. These are the only operations that act on Hilbert space so as to preserve the length of the projection of any one state-vector onto another state-vector.

Consider the parity operator. Acting on the position, it reverses the directions of space, so that PxP−1 = −x. Similarly, it reverses the direction of momentum, so that PpP−1 = −p, where x and p are the position and momentum operators. This preserves the canonical commutator [x, p] = iħ, where ħ is the reduced Planck constant, only if P is chosen to be unitary, PiP−1 = i.

On the other hand, the time reversal operator T, it does nothing to the x-operator, TxT−1 = x, but it reverses the direction of p, so that TpT−1 = −p. The canonical commutator is invariant only if T is chosen to be anti-unitary, i.e., TiT−1 = −i.

Another argument involves energy, the time-component of the momentum. If time reversal were implemented as a unitary operator, it would reverse the sign of the energy just as space-reversal reverses the sign of the momentum. This is not possible, because, unlike momentum, energy is always positive. Since energy in quantum mechanics is defined as the phase factor exp(-iEt) that one gets when one moves forward in time, the way to reverse time while preserving the sign of the energy is to also reverse the sense of "i", so that the sense of phases is reversed.

Similarly, any operation that reverses the sense of phase, which changes the sign of i, will turn positive energies into negative energies unless it also changes the direction of time. So every antiunitary symmetry in a theory with positive energy must reverse the direction of time. Every antiunitary operator can be written as the product of the time reversal operator and a unitary operator that does not reverse time.

This has an interesting consequence on the electric dipole moment (EDM) of any particle. The EDM is defined through the shift in the energy of a state when it is put in an external electric field: Δe = d·E + E·δ·E, where d is called the EDM and δ, the induced dipole moment. One important property of an EDM is that the energy shift due to it changes sign under a parity transformation. However, since d is a vector, its expectation value in a state |ψ> must be proportional to <ψ| J |ψ>, that is the expected spin. Thus, under time reversal, an invariant state must have vanishing EDM. In other words, a non-vanishing EDM signals both P and T symmetry-breaking.[2]

It is interesting to examine this argument further, since one feels that some molecules, such as water, must have EDM irrespective of whether T is a symmetry. This is correct: if a quantum system has degenerate ground states that transform into each other under parity, then time reversal need not be broken to give EDM.

where Φ is a diagonal matrix of phases. As a result, U = ΦUT and UT = UΦ, showing that

U = Φ U Φ.

This means that the entries in Φ are ±1, as a result of which one may have either T2 = ±1. This is specific to the anti-unitarity of T. For a unitary operator, such as the parity, any phase is allowed.

Next, take a Hamiltonian invariant under T. Let |a> and T|a> be two quantum states of the same energy. Now, if T2 = −1, then one finds that the states are orthogonal: a result called Kramers' theorem. This implies that if T2 = −1, then there is a twofold degeneracy in the state. This result in non-relativistic quantum mechanics presages the spin statistics theorem of quantum field theory.

Time reversal transformation for fermions in quantum field theories can be represented by an 8-component spinor in which the above-mentioned T-parity can be a complex number with unit radius. The CPT invariance is not a theorem but a better-to-have property in these class of theories.

Strong measurements (both classical and quantum) are certainly disturbing, causing asymmetry due to second law of thermodynamics. However,
noninvasive measurements should not disturb the evolution so they are expected to be time-symmetric. Surprisingly, it is true only in classical physics but not quantum, even in a thermodynamically invariant equilibrium state.
[1] This type of asymmetry is independent of CPT symmetry but has not yet been confirmed experimentally due to extreme conditions of the checking proposal.